Prove direct product of G1 x G2 is abelian

[kathrynag]

Homework Statement

Prove that if G1 and G2 are abelian, then the direct product G1 x G2 is abelian.

Homework Equations

The Attempt at a Solution

let G1 and G2 be abelian. Then for a1,a2,b1,b2, we have a1b1=b1a1 and a2b2=b2a2.
The direct product is the set of all ordered pairs (x1,x2) such that x1 is in G1 and x2 is in G2.
Let x1 be in G1 and x2 be in G2.
Then G1 x G= (x1, x2)
I'm not quite sure how to show this is abelian.

 

[micromass]

Take two elements in G1 x G2. What does these elements look like? What happens if you multiply them?

 

[kathrynag]

Let a be in G1 b in G2. Then ab=ba

 

[micromass]

No, I take an element in G_1 x G_2. What does this look like?

 

[kathrynag]

oh, so we have (a, b) and (c,d)
We have (a,b)(c,d)=(ac,bd)
But ac=ca and bd=db
So (ac,bd)=(ca,db)=(c,d)(a,b)

 

[micromass]

Yes! good job!

 

[kathrynag]

That's really all I have to do? it seems so simple.

 

[micromass]

Yes, it's that simple